2c09 design for seismic and climate changes · 2014-10-29 · european erasmus mundus master course...
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European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
520121-1-2011-1-CZ-ERA MUNDUS-EMMC
2C09 Design for seismic and climate changes
Lecture 05: Dynamic analysis of multi-degree-of-freedom systems II
Daniel Grecea, Politehnica University of Timisoara
11/03/2014
L6 – Dynamic analysis of multi-degree-of-freedom systems II
European Erasmus Mundus Master Course
Sustainable Constructions under Natural Hazards and Catastrophic Events
L5.1 – Free vibration response. L5.2 – Damping matrix. L5.3 – Modal analysis.
2C09-L5 – Dynamic analysis of multi-degree-of-freedom systems II
Free vibrations of MDOF systems Free vibration of undamped systems
Free vibrations of MDOF systems Free vibration of undamped systems
Free vibrations of MDOF systems Free vibration of undamped systems
Free vibrations of MDOF systems with damping Natural modes of the damped system identical to those of
the undamped system - {}n
Displacements similar to those of the undamped system, but amplitudes decrease with time
Response of each mass is harmonic, similarly to that of a SDOF system
Free vibrations of MDOF systems with damping For each natural mode n, equation of motion in modal
coordinates is
Dividing by Mn one gets: where
The same form as the equation of motion in the case of damped free vibrations of SDOF systems
Combining modal contributions:
0n n n n nM q C q Kq
22 0n n n n n nq q q
2n
nn n
CM
(0) (0)( ) (0)cos sinn nt n n nn n nD nD
nD
q qq t e q t t
21nD n n
1
(0) (0)(0)cos sinn n
Nt n n n
n nD nDnn nD
q qu t e q t t
Modal analysis Equation of motions of a MDOF system with damping
excited by dynamic forces:
Displacements {u} can be expanded as:
Replacing {u} in 4.82:
Multiplying 4.84 to the left by we obtain:
Which, considering orthogonality of natural modes becomes:
m u c u k u p t
1
N
rrr
u q q
4.82
1 1 1
N N N
r r rr r rr r r
m q t c q t k q t p t
4.84
Tn
1 1 1
N N NT T T T
r r rn r n r n r nr r r
m q t c q t k q t p t
n n n n n n nM q t C q t K q t P t 4.86
Modal analysis Modal expansion of displacements
Modal analysis Dynamic response analysis of undamped MDOF system
Modal analysis Dynamic response analysis of undamped MDOF system
Modal analysis Dynamic response analysis of undamped MDOF system
Modal analysis Dynamic response analysis of damped MDOF system
Equation of motions of a MDOF system with damping excited by dynamic forces:
Displacements {u} can be expanded as:
Replacing {u} in 4.82:
Multiplying 4.84 to the left by we obtain:
Which, considering orthogonality of natural modes becomes:
m u c u k u p t
1
N
rrr
u q q
4.82
1 1 1
N N N
r r rr r rr r r
m q t c q t k q t p t
4.84
Tn
1 1 1
N N NT T T T
r r rn r n r n r nr r r
m q t c q t k q t p t
n n n n n n nM q t C q t K q t P t 4.86
Modal analysis Dividing by Mn one gets:
Solving a system of N differential equations was reduced
to solution of N independent equations Direct estimation of the damping matrix [c] not necessary The same form with the equation of motions of a SDOF
system same solution methods Solution: modal coordinate qn(t) for mode n Contribution of mode n to total displacement {u(t)}:
Total displacements (combination of the contribution of
all modes):
22 nn n n n n n
n
P tq q q
M
nnnu t q t
1 1
N N
nnnn n
u t u t q t
Modal analysis Analysis procedure is called modal analysis and is
applicable only to linear systems with classical damping Element forces can be obtained using 2 methods: 1. Contributions rn(t) in n-th mode are obtained from
imposing displacements {u(t)}n Total forces are obtained by superposition of modal contributions
2. Equivalent static forces from the n-th mode are determined: Static analysis modal contributions rn(t) from the n-th mode
1
N
nn
r t r t
2 2n n nnn n n
f t k u t m u t m q t
1
N
nn
r t r t
Modal analysis: summary Define the structural properties
- mass [m] and stiffness [k] matrices - critical damping ratio n
Determine natural circular frequencies n and natural modes of vibrations {}n
Compute response in each mode following the sequence: - set up equation of motion - compute modal displacements {u(t)}n - compute element forces rn(t) from the n-th mode
Combine modal contributions to obtain the total response
Note: generally it is NOT necessary to consider ALL modes of vibration
1 1
N N
nnnn n
u t u t q t
22 nn n n n n n
n
P tq q q
M
Damping matrix Classical damping
Damping matrix Classical damping
Damping matrix Classical damping
Damping matrix Classical damping
References / additional reading Anil Chopra, "Dynamics of Structures: Theory and
Applications to Earthquake Engineering", Prentice-Hall, Upper Saddle River, New Jersey, 2001.
Clough, R.W. şi Penzien, J. (2003). "Dynammics of structures", Third edition, Computers & Structures, Inc., Berkeley, USA
